“Chocolate percentage in the incorrect sauce: 40%.
Chocolate percentage in the pure chocolate: 100%.
Chocolate percentage in the mixture: 50%.
Let I = the incorrect sauce and C = the pure chocolate.
The following approach is called ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS. ...”

“Downhill speed - uphill speed = (x+10) - (x-8) = 18.
Implication:
The downhill speed is 18 mps greater than the uphill speed.
We can PLUG IN THE ANSWERS, which represent the downhill speed.
Since the downhill speed is 18 mps greater than the uphill speed, the correct answer must be GREATER ...”

“On the GMAT, compounded interest is typically just a bit more than simple interest.
Since the answer choices are all greater than 2% -- the percentage given in the prompt -- they must represent the HIGHER interest rate, with 2% representing the LOWER interest rate.
Lower rate:
Since 3/4 of ...”

“Rule:
MULTIPLE OF X + MULTIPLE OF X = MULTIPLE OF X
MULTIPLE OF X + NON-MULTIPLE OF X = NON-MULTIPLE OF X
225 = 9*25
Statements combined:
Case 1: m=9 and n=25*15, with the result that m is a multiple of 9, n is a multiple of 15, and mn = 9*25*15 = 225*15 = multiple of 225
In this case, m+n ...”

“A fraction will yield a terminating decimal if the prime-factorization of its denominator contains only 2''s and/or 5''s.
n/420 = n/(2*2*3*5*7).
For a terminating decimal to be yielded, the prime factors in red must CANCEL OUT with n, leaving only 2''s and 5''s in the denominator.
Thus, the ...”

“Generally, a one-word adverb such as tirelessly will not be positioned between a verb and its direct object.
Incorrect: Companies lobby tirelessly government officials.
Here, tirelessly (adverb) is awkwardly positioned between lobby (verb) and government officials (direct object).
Correct: ...”

“Some grammarians -- including me -- reserve each other for two elements and one another for more than two.
Other grammarians consider the two modifiers interchangeable.
A club has exactly three members: Ann, Beth, and Carol.
The sentence above implies the following:
Ann, Beth and Carol ...”

“An INSCRIBED ANGLE is formed by two chords.
Thus, angle PQR is an inscribed angle.
An inscribed angle that intercepts the diameter is a RIGHT ANGLE.
Thus, angle PQR is a right angle, implying that triangle PQR is a RIGHT TRIANGLE.
http://s15.postimage.org/xozmfjdh3/PQR.jpg
In the figure ...”

“Test the THRESHOLD.
Since we want to know whether the speed from A to B is greater than 40mph, the threshold here is 40mph.
Statement 1: Reiko''s average speed for the entire round trip, excluding the time spent at point B, was 80 miles per hour.
Let the distance in each direction = 40 ...”

“Since a small doll costs $1 less than a large doll, the answer choices imply the following options for the price of a small doll:
4
5
6.2
7
8
Given that all of the values in the prompt are INTEGERS, the price of a small doll is almost certain to be a factor of 360.
Only the values in blue ...”

“Determine the WORST-CASE-SCENARIO -- for each color of ball, the MAXIMUM number that can be removed WITHOUT removing 5 of the same color:
3 red
4 green
4 yellow
4 blue
4 white
Sum = 3+4+4+4+4 = 19
Implication:
It is possible to remove 19 balls without selecting 5 of the same color.
...”

“Prompt:
Total production = 1000n
Statement 1:
Total production for the first 5 days = 5*600 = 3000
Statement 2:
Total production for the last 4 days = 4*1500 = 6000
Case 1: n=9, with no days between the first 5 days and the last 4 days
Total production = 1000n = 1000*9 = 9000
Total ...”

“What is the remainder when (n+1)(n-1) is divided by 8?
Statement 1:
Since n is odd, both n-1 and n+1 are EVEN, implying that (n-1)(n+1) = the product of two consecutive even integers.
Given any two consecutive even integers, one will be a multiple of 4, while the other will be an even ...”

“Always look for special triangles such as 3-4-5 and 5-12-13.
Statement 2:
Case 1:
https://i.postimg.cc/V0JX8Hqv/rectangle-with-x-y-and-z-1.png
Case 2:
https://i.postimg.cc/DWCWwPyX/rectangle-with-x-y-and-z-2.png
Since each case will yield a different perimeter, INSUFFICIENT.
Statement ...”

“The portion in red is incorrect: it should be not 4C1 but 2C1.
If we select two upper case letters from two sets of twins, we must then select one lower case letter from the OTHER TWO SETS OF TWINS, yielding only two options for the lower case letter.
For example:
If we select A and B for the two ...”

“Statement 1:
Any integer whose digits sum to a multiple of 3 must itself be a multiple of 3.
Since N has a digit sum of 11, N+1 must have a digit sum of 12.
One exception: If N=29, then N+1 = 30, which has a digit sum of 3.
Since the digit sum of N+1 must be a multiple of 3 -- either 12 or 3 -- ...”

“Statement 1:
Since (49, 0) and (-49, 0) are both solutions, the quadratic must be as follows:
f(x) = (x-49)(x+49) = x² - 49²
In the resulting quadratic, b=0.
Thus, the answer to the question stem is YES.
SUFFICIENT.
Statement 2:
If x-49 is a factor of the equation, then f(49) = 0.
Case ...”

“Range = biggest - smallest
Smallest = biggest - range
Since the range = 60, we get:
Smallest = biggest - 60
To minimize the smallest integer, we must minimize the biggest integer.
Since the median of the 51 distinct integers = 30, there must be 25 integers above 30, with the other 25 ...”

“To form a diagonal, we must choose 2 vertices.
But any pair of vertices that forms a side of the polygon cannot serve to form a diagonal.
Thus:
Number of diagonals = (number of ways to choose 2 vertices) - (number of sides of the polygon)
When the prompt includes the phrase which of the ...”

“To count circular arrangements:
1. Place someone in the circle
2. Count the number of ways to arrange the REMAINING people
Once Alice has been placed in the circle:
Number of options for Bob = 1. (Must be in the seat opposite Alice)
Number of options for Cindy = 4. (Any of the 4 remaining ...”

“E: mosaic designs with which artisans who lived in various parts of the Roman Empire were familiar.
Here, the two blue portions constitute an adjective serving to modify designs.
Question: What KIND of designs?
Answer: Designs WITH WHICH ARTISANS WERE FAMILIAR.
Put another way:
Artisans were ...”

“Rate and time have a RECIPROCAL relationship.
If the hourly rate DOUBLES, then the same income will be yielded in 1/2 the total time.
Statement 1:
Actual hourly rate = 225/30 = $7.50, implying that an additional $1.50 per hour will increase the hourly rate to $9.
(increased hourly ...”

“M --> more than 50% of all disease cases over the past 5 years
S --> 40% of all disease cases 5 years ago, 30% of all disease cases now
For each answer choice:
Try to devise a case that aligns with the information in the passage but CONTRADICTS the answer choice, thus proving that the ...”

“Generally, COMMA + SINCE after a clause serves to express a CAUSAL relationship.
An OA in GMATPrep:
Prospecting for gold was a relatively easy task, since erosion put gold literally within reach of anybody with a pan.
Here, the clause in blue serves to express what CAUSED prospecting to be a ...”

“Statements combined:
Case 1: p=11 --> p is prime, but its units digits is not prime
In this case, 2p+1 = 2*11 + 1 = 23.
Since 23 is prime, the answer to the question stem is YES.
Case 2: p=19 --> p is prime, but its units digit is not prime
In this case, 2p+1 = 2*19 + 1 = 39.
Since 39 ...”

“When one positive integer is divided by another, we typically represent what is left over as a REMAINDER or as a DECIMAL.
There is a relationship between the two representations:
Remainder/Divisor = Decimal.
When 5 is divided by 2:
Remainder representation: 5/2 = 2 R1
Decimal ...”

“Let y=$10 and R=10 students, with the result that the original share per student = ($10)/(10 students) = $1.
Let x=9, implying that the number of students decreases by 9 to 1.
New share per student = ($10)/(1 student) = $10.
Increase in share per student = 10-1 = $9.
The correct answer must ...”

“Statement 1:
In other words, y is HALFWAY BETWEEN x and z, implying that x, y and z are EVENLY SPACED.
For any evenly spaced set, the average = the median.
Thus, the answer to the question stem is YES.
SUFFICIENT.
Statement 2:
x+z = 2y
(x+z)/2 = y
Implication:
y is the average of x and z ...”

“This problem is not relevant to the GMAT.
On the GMAT, problems involving remainders are constrained to POSITIVE INTEGERS.
From the Math Review in the OG:
If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y = xq + r ...”

“Given that the area and the answer choices are all integers, the triangle is probably a Pythagorean Triple.
The only Pythagorean Triple with an area of 30 is 5-12-13.
In the given equation, test c=5, with the result that c+7 = 12:
x/5 + y/12 - 1 = 0
If x=0, then y=12, implying a y-intercept ...”

“Dividing 32 by k leaves a remainder of k-3.
In other words, 32 is equal to k-3 more than a multiple of k:
32 = ak + (k-3), where a is a nonnegative integer and k is an integer such that k≥3.
Simplifying 32 = ak + (k-3), we get:
35 = ak + k
35 = k(a+1)
The resulting equation implies that ...”

“Identify two perfect squares that sum to 25:
1, 4, 9, 16...
The two values in blue sum to 25.
If y=3 and x=7, then (x-y)² = (7-3)² = 16 and y² = 9, with the result that (x-y)² + y² = 25.
The correct answer is B.”

“Generally, COMMA + VERBing serves to modify the nearest preceding action and the agent of this action.
Given this rule, I received a PM requesting that I justify the usage of COMMA + needing in the OA:
Insurance companies in California were free to charge whatever rates the market would bear, ...”

“π ≈ 3.
Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.
Circumference of track = 20π ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now ...”

“The two factors must have DIFFERENT SIGNS.
Each of the answer choices will yield a positive value for x² + 6x + 10.
Thus, the correct answer must yield a negative value for 2x-49.
Only A is viable:
(2*24) - 49 = -1.
The correct answer is A.”

“Option D indicates that 12 pairs of cheap socks can be bought, not 9.
Unfortunately, 9 is not among the answer choices.
The OA implies that each pair of cheap socks costs not $2 but $1.50.
Because $1.50 is not an integer value, the brute-force approach is a bit tricky here.”

“We can PLUG IN THE ANSWERS, which represent the number of cheap pairs that can be purchased for $18.
When the correct answer is plugged in, the number of expensive pairs that can be purchased will be 3 less than the number of cheap pairs that can be purchased.
B will yield an ugly price for each ...”

“Use the following equation:
(machines)(time) / output = (machines)(time) / output
In the equation above:
Machines and time are INVERSELY PROPORTIONAL.
As the number of machines increases, the amount of time required to produce the same output decreases.
Machines and output are DIRECTLY ...”

“Multiplying the equation by -1, we get:
y = x² - 16x - 36
y = (x-18)(x+2)
The restorative power -- the value of y -- is equal to 0 when x=18 or x=-2.
Since the time cannot be a negative value, only x = 18 hours is viable.
The correct answer is E.”

“What is (June X)/(June Y)?
Statement 1: In June 1989, Salesperson X made 50 percent more sales transactions than Salesperson Y did in May 1989.
(June X)/(May Y) = 150/100 = 3/2.
INSUFFICIENT.
Statement 2: In June 1989, Salesperson Y made 25 percent more sales transactions than in May ...”

“(integers with the digit 2) = (all integers between 100 and 399, inclusive) - (integers without the digit 2)
All integers between 100 and 399, inclusive:
For any set of consecutive integers, count = biggest - smallest + 1.
In this case:
count = 399-100+1 = 300
Integers without the digit 2:
...”

“We can PLUG IN THE ANSWERS, which represent a side of the square.
Since the dimension of the rectangular playground are INTEGER values, the correct answer is probably also an integer.
When the correct answer is plugged in, the square and the rectangle will have the same area.
D: 50, implying ...”

“If 10 rooms are rented for $40 instead of $60 -- a decrease of $20 per room -- the total decrease = 10*20 = $200.
Since the value in blue constitutes a 25% reduction in the total rent, it must be equal to 1/4 of the total rent:
200 = (1/4)R
800 = R.
The correct answer is B.”

“How many umbrellas can it afford to sell at the below-cost rate of b dollars per umbrella and still make a 100% profit?
If c=1 and r=2 -- implying that the cost per umbrella = $1 and that the retail price per umbrella = $2 -- then 100% profit will be yielded only if ALL of the umbrellas are sold at ...”

“We can PLUG IN THE ANSWERS, which represent the number of rainy days.
When the correct answer is plugged in, the total number of cups = 26.
B: 3 rainy days, implying 4 non-rainy days
Since 5 cups of tea are consumed are on each of the 4 non-rainy days, the total number of tea cups = 5*4 = 20.
...”

“(new rate)/(old rate) = (6 miles per hour)/(22 miles per hour) = 6/22 = 3/11.
Rate and time have a RECIPROCAL relationship.
Since the new rate is 3/11 of the old rate, the new time must be 11/3 of the old time:
(11/3)(9 minutes) = 33 minutes.
The correct answer is B.”

“The GMAT is unlikely to test divisibility by 7.
An integer of the form XYZXYZ must be divisible by 7, but this issue seems irrelevant to the GMAT.
For this reason, I''ve replaced answer choice A with the value in red:
To determine whether an integer is divisible by 11:
1. From the left to ...”

“For any set of consecutive integers:
count = biggest - smallest + 1
Thus:
Number of two-digit integers between 10 and 99, inclusive = 99-10+1 = 90.
Of these 90 consecutive integers, 1 of every 3 will be a multiple of 3, implying that 2/3 will NOT be divisible by 3:
(2/3)(90) = 60.
Of these ...”

“Why do people who do regular volunteer work live longer?
The argument concludes that ENDORPHINS are the cause.
One way to weaken this conclusion is to suggest an ALTERNATE CAUSE.
D: People tend not to become involved in regular volunteer work unless they are healthy and energetic to begin with. ...”

“When the SAME DISTANCE is traveled at two different speed, the average speed for the entire trip will be just a little less than the average of the two speeds.
The reason:
The trip at the lower speed takes longer than the trip at the higher speed.
Since more time is spent at the lower speed, the ...”

“To MINIMIZE the percent for the largest balance, MAXIMIZE the percents for the 3 smallest balances.
We can PLUG IN THE ANSWERS, which represent the percent in the largest balance.
When the correct answer is plugged in, the 4 percents will sum to 100%.
Since the question stem asks for the smallest ...”

“Let N dollars in 1992 = 1.
Since the assets in 1993 are 300% greater than the N dollars in 1992, the assets in 1993 = 1 + (300% of 1) = 1+3 = 4.
Since the assets in 1994 are 400% greater than the assets in 1993, the assets in 1994 = 4 + (400% of 4) = 4 + 16 = 20.
The question stem asks for ...”

“(50 days)/(60 days) = 5/6.
To finish the job in 5/6 of the normal time, the crew requires 6/5 of the normal number of students -- an increase of 1/5:
(1/5)(10) = 2 more students.
The correct answer is A.”

“From 7 batsmen, the number of ways to choose 5 = 7C5 = (7*6*5*4*3)/(5*4*3*2*1) = 21.
From 6 bowlers, the number of ways to choose 4 = 6C4 = (6*5*4*3)/(4*3*2*1) = 15.
From 3 wicketkeepers, the number of ways to choose 1 = 3.
From 3 allrounders, the number of ways to choose 1 = 3.
To combine these ...”

“Let x=1 and y=3/2, with the result that 1/z = 1 - 2/3 = 1/3, so z=3.
Since the question stem asks for the value of y, the correct answer must yield 3/2 when x=1 and z=3.
Since B and D include x-z and thus will yield a negative result, eliminate B and D.
Since E includes only multiplication and ...”

“Statements combined:
Case 1: p=0, with the result that p is an integer less than 1
In this case, p³=p, so the answer to the question stem is NO.
Case 2: p=-2, with the result that p is an integer less than 1
In this case, p³ < p, so the answer to the question stem is YES.
Since the answer ...”

“https://i.postimg.cc/06HLPxqC/circle-in-a-sector.png
AOP passes through the center of the inscribed circle and thus must BISECT sector ACB.
AB, AP and AC are all radii of sector ACB and thus are equal.
The prompt indicates AB=AC=12.
Thus:
AP=12.
As shown in the figure, AOM is a 30-60-90 ...”

“25% greater = 1/4 greater.
If the smaller integer = 4, then the larger integer = 4 + (1/4)4 = 5.
The values in red imply that the two integers summing to 27 must be in a RATIO OF 4 TO 5:
4, 5
8, 10
12, 15
The case in blue sums to 27, with the result that larger-smaller = 15-12 = 3.
The ...”

“Since b and c form a straight line, b+c = 180.
Thus, the calculate a+b+c, we need to know the value of a.
Question stem, rephrased:
What is the value of a?
Statement 1:
SUFFICIENT.
Statement 2:
No information about a.
INSUFFICIENT.
The correct answer is A.”

“Total number of items brought to the meeting = 21. (10 charts and 11 pencils)
Number of items brought by each person = 3. (Either 2 pencils and 1 chart or 1 pencil and 2 charts)
Thus:
Number of people = (total number of items)/(number of items per person) = 21/3 = 7.
The correct answer is B.”

“Determine the WORST-CASE-SCENARIO -- for each color of ball, the MAXIMUM number that can be removed WITHOUT removing 7 of the same color:
1 red
3 green
5 yellow
6 blue
6 white
6 black
Sum = 1+3+5+6+6+6 = 27.
Implication:
It is possible to remove 27 balls without selecting 7 of the same ...”

“The system will have an infinite number of solutions for (x, y) if the two equations are THE SAME:
2(x+y) - A = 0 --> 2x + 2y - A = 0 --> 6x + 6y - 3A = 0
3x + By - 6 = 0 -----------------------> 6x +2By - 12 = 0
The equations in blue will be the same if 3A=12 and 2B=6, with the result ...”

“When two elements travel at different speeds, their TIME RATIO to travel the same distance will always be the same.
If A takes 1/2 as long as B to travel 10 miles, then A will take 1/2 as long as B to travel 1000 miles.
If A takes 3 times as long as B to travel 500 miles, then A will take three ...”

“Let the 5 colors be GYABC.
From the 5 colors, the number of ways to choose 3 = 5C3 = (5*4*3)/(3*2*1) = 10.
Of these 10 combinations, 3 are not allowed:
GYA
GYB
GYC
Subtracting these 3 bad combinations, we get:
10-3 = 7.
The correct answer is A.”

“MULTIPLE OF X + MULTIPLE OF X = MULTIPLE OF X.
MULTIPLE OF X + NON-MULTIPLE OF X = NON-MULTIPLE OF X.
Since the prime-factorization of 15! includes 13, we get:
a = 15! + 13 = multiple of 13 + multiple of 13 = multiple of 13.
Since a is a multiple of 13, option I is a factor of a.
Eliminate ...”

“AND means MULTIPLY.
OR means ADD.
Case 1: 2 doves
P(1st animal is a dove) = 3/5. (Of the 5 animals, 3 are doves.)
P(2nd animal is a dove) = 2/4. (Of the 4 remaining animals, 2 are doves.)
Since in Case 1 we need the 1st animal to be a dove AND the 2nd animal to be a dove, we MULTIPLY the ...”

“Statement 1:
X = 20 miles per gallon.
Y = 40 miles per gallon.
Since the mixture of X and Y in the tank allows 200 miles to be covered using 8 gallons, we get:
X+Y = 200/8 = 25 miles per gallon.
To determine how X and Y must be combined to yield a rate of 25 miles per gallon, use ALLIGATION.
...”

“To make the math easier, we can replace the given values with rounder numbers, as follows:
Since Rebeca arrives 1 hour late when traveling at the lower speed and 1 hour early when traveling at the higher speed, the time at the lower speed must be 2 hours greater than the time at the higher ...”

“100(11 - √119)(11 + √119) = 100(11² - √119²) = 100(121-119) = 100*2 = 200.
When the correct answer is multiplied by 11+√119, the result must be as close as possible to 200.
11+√119 = 11 + (a bit less than 11) = a bit less than 22.
Since 9*22 = 198, multiplying 9.2 by a bit less than ...”

“Let x=12.
Since 1/4 of the people rise up, the number who rise up = (1/4)(12) = 3.
Since 3 people come back down, EVERYONE who rises up comes back down, with the result that all 12 people are practicing yoga on the ground.
How many people are now practicing yoga on the ground?
The correct ...”

“Statement combined:
Case 1: X = 53*2 = 106, with the result that X is divisible by 1, 2, 53, and 106
In this case, X has TWO prime factors:
2 and 53
Case 2: X = 53³, with the result that X is divisible by 1, 53, 53², and 53³
In this case, X has only ONE prime factor:
53
Since the number of ...”

“A fraction will yield a terminating decimal if -- when the fraction is fully reduced -- the prime-factorization of its denominator includes only 2''s and or 5''s.
Statement 1:
Let x = an odd multiple of 5.
Since the next largest odd multiples of 5 will be x+10 and x+20, we get:
p = x + (x+10) ...”

“A radius drawn to a tangent point yields a RIGHT ANGLE:
https://i.postimg.cc/Wq6BDNH7/two-triangles-and-semicircle-1.png
In the figure above:
Radius OT forms a right angle at tangent point T.
OA and OT are radii and thus are equal.
Since OA=OT, opposite angles OAT and ATO are equal.
We can ...”

“Total number of friendly numbers = total number of ways to arrange the 5 digits = 5! = 120.
There are 5 options for the first digit: 1, 2, 3, 4, or 5.
120/5 = 24.
Thus, the 120 friendly numbers are composed of five 24-number ranges, as follows:
The integers beginning with 1 constitute friendly ...”

“An integer is divisible by 4 if its last two digits form a multiple of 4.
Two-digit multiples of 4 with a units digit of 2:
12, 32, 52, 72, 92
Since the last two digits must be one of the options above, between 1 and 200 there are 10 integers that have a units digit of 2 and are divisible by 4:
...”

“Options D and E should appear as shown below:
P(exactly n times) = P(one way) * all possible ways.
Let R = rain and N = no rain.
Since P(R) = 20% = 2/10, P(N) = 80% = 8/10 = 2³/10.
P(one way):
One way to get exactly 2 days of rain is to have rain on the first 2 days but not on the ...”

“The sides of an isosceles right triangle are in the following ratio:
x - x - x√2, where x and x constitute the base and height and x√2 is the hypotenuse.
Since the triangle has a base of x, a height of x, and an area of 16, we get:
(1/2)(x)(x) = 16
x² = 32
x = √32
Since XY is longer ...”

“Statement 1:
Time and rate are RECIPROCALS.
Since X and Y together take 2/3 as long as X alone, X and Y together work 3/2 as fast as X alone.
Thus, if X''s rate alone = 2 units per hour, then X and Y''s combined rate = (3/2)(2) = 3 units per hour, implying that Y''s rate alone = 3-2 = 1 unit ...”

“Since A and B must occupy adjacent positions, consider AB a single element in the arrangement.
The number of ways to arrange the 6 elements AB, C, D, E, F and G = 6! = 720.
In 1/2 of these arrangements, C will be to the LEFT of AB.
In the remaining 1/2 of these arrangements,C will be to the ...”

“Nice problem, Fabio!
Statement 1:
Case 1: a=4, b=1 and c=1, with the result that b+c≥2
In this case, abc = 4, so the answer to the question stem is YES.
Case 2: a=0, b=1 and c=1, with the result that b+c≥2
In this case, abc = 0, so the answer to the question stem is NO.
Since the answer ...”

“Statement 1:
If n=5, then the remainder when n is divided by 6 = 5/6 = 0 R5.
If n=10, then the remainder when n is divided by 6 = 10/6 = 1 R4.
Since different remainders are possible, INSUFFICIENT.
Statement 2:
Since n is a multiple of 12, n must also be a multiple of 6, with the result that ...”

“Statement 1: The product of the x-intercepts of line L and K is positive.
K has a negative x-intercept, L has a negative x-intercept, the product of the slopes is positive:
http://s4.postimage.org/ie5nbe78/lines_l_and_k_1.jpg
K has a positive x-intercept, L has a positive x-intercept, the ...”

“Statements combined:
Case 1: x=1 and y=1, with the result that x(y+5) = 6 and 6y²+41y+25 = 72
In this case, x is ODD, so the answer to the question stem is NO.
Case 2: x=2 and y=1, with the result that x(y+5) = 12 and 6y²+41y+25 = 72
In this case, x is EVEN, so the answer to the question ...”

“a+b+3 = ODD only if a and b are both odd or both even.
Question stem, rephrased:
Are a and b both odd or both even?
Statement 1:
ab = ODD implies that a and b are both odd.
Thus, the answer to the question stem is YES.
SUFFICIENT.
Statement 2:
a-b = EVEN implies that a and b are either ...”

“Alternate approach:
Case 1: VVCC, where the first two letters are vowels and the last 2 letters are consonants
Number of options for the first vowel = 4. (Any of the 4 vowels)
Number of options for the second vowel = 3. (Any of the 3 remaining vowels)
Number of options for the first consonant ...”

“x = 10% gravel
y = 2% gravel.
z = the MIXTURE of x and y = 5% gravel.
To determine the required ratio of x to y, use ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.
Step 1: Plot the 3 percentages on a number line, with the percentages for x and y on the ends and the ...”

“I received a PM requesting that I respond.
Premise:
Insurance removes from depositors any financial incentive to find out whether the bank that holds their money is secure against failure.
Conclusion:
The insurance is partly responsible for the high rate of bank failures.
To determine the ...”

“Every student is EITHER male OR female.
Every student is EITER older OR younger.
For an EITHER/OR problem, use a DOUBLE MATRIX to organize the data:
_______________M______F_______Total
older:
younger:
total:
Let total students = 100:
_______________M______F_______Total
...”

“Let y=0, implying that NONE of the 80 houses are painted at the original rate of x houses per week, with the result that ALL 80 houses are painted at the greater rate of (5/4)x houses per week.
The total time it takes them to paint all the houses under this scenario is what fraction of the time ...”

“Since J catches up to K, J and K travel the same distance.
Let t = J''s time.
Since K leaves 15 minutes earlier than J, K''s time = t+15.
Rate and time have a RECIPROCAL RELATIONSHIP.
Since K''s rate is 80% of J''s rate, the ratio of K''s rate to J''s rate = 8/10 = 4/5.
Thus, the ratio of ...”

“(students per section)(number of sections) = 595 students
number of sections = 595/(students per section).
The equation in blue indicates that 595 must be divisible by the number of students per section.
An integer is divisible by 3 only if its digit sum is a multiple of 3.
The digit sum of ...”

“Correct: The contractors demolished the building to keep it from falling.
Here, to keep is not an adjective modifying contractors but an ADVERB modifying demolished, expressing WHY the contractors DEMOLISHED the building.
Question: WHY did the contractors demolish the building?
Answer: TO KEEP it ...”

“Let the PRE-TOWER number of beds = the LCM of the denominators in the problem = 3*5 = 15.
Then the CURRENT number of beds = 3*15 = 45, implying that the number of beds in the NEW tower = 45-15 = 30.
Occupied beds = (1/3)15 + (1/5)30 = 11.
Non-occupied beds in the new tower = 30-11 = 19.
...”

“One more approach:
Since A+B together complete the job in t minutes:
Job = (A''s output in t minutes) + (B''s output in t minutes)
If A works alone for t minutes, A can then complete the rest of the job in 64 minutes.
In this case, the rest of the job = B''s output in t minutes.
Thus, A''s ...”

“A present perfect verb (has + VERBed or have + VERBed) expresses a PAST ACTION that affects the present.
Generally, the usage of a present perfect verb implies that the subject of the verb still exists.
Incorrect:
The dinosaurs have evolved over time.
The sentence above is illogical because the ...”

“Always keep your eye on the answer choices.
Since A''s time alone (t+64) is greater than B''s time alone (t+25), the ratio of A''s time alone to B''s time alone must be GREATER THAN 1.
Eliminate A, C and D.
Since the ratio of A''s time to B''s time = (t+64)/(t+25), option E implies the ...”

“First day:
Total fine = 10.
For each additional day, the total fine either increases by 30 or doubles -- whichever results in a lesser amount.
Second day:
Adding 30 cents to the 1-day fine, we get 10+30 = 40.
Doubling the 1-day fine, we get 2*10 = 20.
Total fine = lesser amount = 20. ...”

“Clearly, neither statement alone is sufficient.
Statements combined:
Case 1: No one is a member of both clubs
Total males = 15+10 = 25.
Total members = 16+20 = 36.
Resulting fraction:
(total males)/(total members) = 25/36.
Case 2: Every male in the music club is also in the drama ...”

“Let d = the number of miles between A and B.
Since Georgia travels the entire d miles between A and B and then an additional 60 miles back toward A, Georgia''s distance = d+60.
Since Frank travels all but 60 of the d miles between A and B, Frank''s distance = d-60.
Since Georgia''s travels 3/2 ...”

“Let S = the entire season (in other words, the mixture of M and N).
Alligation can be performed only with percentages or fractions.
Step 1: Convert the ratios to FRACTIONS.
M:
Since win:losses = 1:2, and 1+2=3, wins/total = 1/3.
N:
Since wins:losses = 1:3, and 1+3=4, wins/total = 1/4.
...”

“Since Stacy and Katie travel toward each other, they WORK TOGETHER to cover the 27 miles between them.
Since Stacy is 1/4 faster than Katie, for every 4 miles that Katie travels, the distance traveled by Stacy = 4 + (1/4)(4) = 5 miles.
Thus, of every 9 miles traveled when Stacy and Katie work ...”

“Since Mary is behind Kate by 250 meters -- half the circumference of the track -- she will catch up to Mary when she has traveled 250 meters more than Kate.
To catch up to Kate a second time, Mary must then travel one more lap than Kate -- in other words, 500 meters more than Kate.
Implication:
...”

“|x-1| = the distance between x and 1.
|x+1| = the distance between x and -1.
-1<--|x+1|-->x<--|x-1|-->1
Here, since the total distance between the two endpoints is 2, the sum of the 2 distances in red is 2.
If x is any value between the two endpoints, then the sum of the two ...”

“were + VERBing is the plural form of the PAST PROGRESSIVE tense (also known as the past continuous tense).
The past progressive tenses serve to express actions that happen CONTINUOUSLY FOR A LIMITED DURATION OF TIME.
OA: While businesses were expanding their production, unsold goods were piling ...”

“To minimize B''s time, we must maximize A''s time.
Since each machine can work for at most 8 hours, let A''s time = 8 hours.
Since A''s rate = 9000 pencils per hour, the number of pencils produced by A in 8 hours = rt = 9000*8 = 72,000.
Remaining pencils = 100,000 - 72,000, = 28,000.
Since ...”

“Test the THRESHOLD.
In the question stem, the threshold is 40 CHAIRS.
The total cost of 40 chairs = 40*50 = 2000.
Statement 1: The total price of the desks and chairs is 5,000.
It is possible that 40 chairs are purchased (at a total cost of $2000) or that MORE CHAIRS are purchased.
...”

“Monthly food expense except in January:
Let y=$2.
Discount in January:
Let x=100%.
Since $2 is spent each month for 8 of the first 9 months, the total spent during these 8 months = 8*2 = 16.
Since food purchases are 100% lower in January -- implying that they are FREE in January -- the amount ...”

“Alternate approach:
To determine the ratio of birds to dogs, plug in a TEST VALUE for the total number of pets.
Let b = the number of birds and d = the number of dogs.
Let the test value for the total number of pets = the LCM of 3 and 5 = 15.
Since 1/5 are birds, b = 1/5 * 15 = 3.
Since 1/3 ...”

“Statement 1:
If d=72, then t = d/r = 72/72 = 1 hour.
If d=144, then t = 144/72 = 2 hours.
Since the time can be different values, INSUFFICIENT.
Statement 2:
Time and rate have a RECIPROCAL RELATIONSHIP:
If the time is cut in HALF, then the rate DOUBLES.
If the time DOUBLES, then the ...”

“Let G = the gas in the tank and C = the capacity of the tank.
Statement 1:
G - (1/2)G = (1/3)C
No way to solve for G.
INSUFFICENT.
Statement 2:
G + 0.75 = C
No way to solve for G.
INSUFFICIENT.
Statements combined:
Since we have two variables (G and C) and two distinct linear ...”

“Here, the value in red is the sum of Jake''s weight and TWICE the sister''s weight:
187 + 2(89.5) = 366.
But the prompt gives an integer value for the sum of Jake''s weight and the sister''s ACTUAL weight -- a constraint that forces Jake''s weight to be an even integer.”

“The sum of Jake''s weight and his sister''s weight is an INTEGER VALUE.
The answer choices indicate that Jake''s weight is an INTEGER VALUE.
Implication:
The sister''s weight must also be an integer value.
Thus, the value in red -- 89.5 -- is not a valid option for the sister''s weight.
Given ...”

“Let the river''s rate = 1 mph and the boat''s rate = 5 mph.
When the boat travels DOWNRIVER, the boat and the river WORK TOGETHER, so we ADD their rates:
5+1 = 6 mph.
When the boat travels UPRIVER, the boat and the river COMPETE, so we SUBTRACT their rates:
5-1 = 4 mph.
Rate and time have ...”

“ALWAYS KEEP YOUR EYE ON THE ANSWER CHOICES.
If Jake loses 8 pounds, he will weigh twice as much as his sister.
j-8 = 2s
j = 2s + 8 = even + even = even.
Since Jake''s weight must be an EVEN VALUE, the correct answer is E.”

“Let S = Sara''s age.
Sum of the ages = 2+4+5+8+10+13+S = 42+S.
Statement 1:
Since the average age is equal to an INTEGER -- the fourth largest age -- we get:
Sum of the ages = (number of ages)(average age) = (7)(INTEGER) = multiple of 7.
The blue expression and the red expression each ...”

“We can PLUG IN THE ANSWERS, which represent the value of m.
When the correct answer is plugged in, the median will be equal to the range.
Note:
When the prompt includes the phrase which of the following, the correct answer is likely to be D or E.
E: m=3
Resulting set: 1, 1, 2, 2, m=3, 3, 3, ...”

“Let X=8 and Y=4.
Henry eats X scones in X percent of the time it takes Rachel to eat Y scones.
Since Henry eats X=8 scones in 8% of the time it takes Rachel to eat Y=4 scones -- 10 minutes, according to the prompt -- the time for Henry to eat 8 scones = 8% of 10 minutes = (8/100)(10) = 4/5 ...”

“Consider the following equation:
7x + 5y = 70.
If x and y are nonnegative integers, the following solutions are possible:
x=10, y=0
x=5, y=7
x=0, y=14.
Notice the following:
The value of x changes in increments of 5 (the coefficient for y).
The value of y changes in increments of ...”

“x is directly proportional to y implies the following:
x/y = k.
x = ky.
In each case, k is a CONSTANT.
The cost of a square slab is proportional to its thickness and also proportional to the square of its length.
Thus:
c = k * T * L².
What is the cost of a square slab that is 3 meters ...”

“species may be singular or plural.
Passage:
Planting trees primarily from the species that turns out to have the best record of survival will assure downtown Thomasville of an abundant tree population 20 years from now.
Here, the verb attributed to species is turns.
Since turns is singular, ...”

“(number of machines)(number of hours) = number of toys that are produced.
In the prompt above, the blue values are THE SAME.
Implication:
2 machines take h+3 hours to produce the same number of toys as 3 machines in 2 hours:
2(h+3) = 3h
2h + 6 = 3h
6 = h.
Since 3 machines produce 360 toys ...”

“Let x = 5 seconds for the first lap, implying 10 seconds for the second lap, 3 seconds for the third lap, and a total of 18 seconds for all 3 laps.
(18 seconds)/(3 laps) = (6 seconds)/(1 lap) = 1/10 minute per lap.
The correct answer must yield a value of 1/10 when x=5.
Only D works:
(x+1)/60 ...”

“The information in red is irrelevant.
Only the following facts matter:
Ross travels onward at a rate of 1 mile every 12 minutes.
He travels home for twice the onward distance at half the onward speed.
Onward speed = 1 mile per 12 minutes = 5 miles per 60 minutes = 5 mph.
Let the onward ...”

“The original total cost of the 10 pieces of fruit = 10*56 = 560.
According to the answers, after 1, 2, 3, 4, or 5 oranges are removed -- so that 9, 8, 7, 6, or 5 pieces of fruit remain -- the average cost per piece decreases to 52.
Since the original total cost is a multiple of 10, and the price ...”

“Let x = the product of the two denominators in the prompt = 3*4 = 12.
Since 1/3 won a medal for swimming, the number who won a medal for swimming = (1/3)(12) = 4.
Since 1/4 of the medal winners for swimming also won a medal for diving, the number who won both a medal for swimming and a medal for ...”

“Given two sides of a triangle, the greatest possible area will be achieved if the two sides form a RIGHT ANGLE and serve as the triangle''s base and height.
For the problem above, I posted an illustrated solution here:
...”

“When the SAME DISTANCE is traveled at two different speeds, the average speed for the entire trip will be CLOSER TO THE LOWER SPEED than to the higher speed.
The reason:
The portion traveled at the lower speed takes LONGER.
Since more time is spent at the lower speed than at the higher speed, the ...”

“Statement 1: The numbers of pens, pencils, and pads that each staff member received were in the ratio 2:3:4, respectively.
Since 2+3+4 = 9, the total number of pens + pencils + pads must be a multiple of 9.
No way to determine the exact number of staff members.
INSUFFICIENT.
Statement 2: ...”

“Both of the following are viable:
Vehicles powered by natural gas emit less of certain pollutants than those burning gasoline or diesel fuel.
Vehicles powered by ndatural gas emit less of certain pollutants than do those burning gasoline or diesel fuel.
The inclusion of do is optional.
With or ...”

“What is the value of n-m?
(m,0) and (n,0) are the two points where y=0 and the graph intersects the x-axis.
Thus, m and n are the x-intercepts of the graph.
Since m < n, n-m > 0.
Question rephrased: What is the positive difference between the x-intercepts?
Statement 1: 4b = a²-4
Test ...”

“percent may serve as a NOUN.
Of the students in the class, eighty percent passed the test.
Here, eighty percent is a NOUN serving as the subject of passed.
twice is a MODIFIER.
It cannot serve as the subject of a sentence.
Of the students in the class, twice passed the test.
The sentence ...”

“Consider the following equation:
5x + 7y = 70.
If x and y are nonnegative integers, the following solutions are possible:
x=0, y=10
x=7, y=5
x=14, y=0.
Notice the following:
The value of x changes in increments of 7 (the coefficient for y).
The value of y changes in increments of 5 (the ...”

“The ratio of milk and water in a mixture is 2:1,
Let milk = 2 liters and water = 1 liter.
Some quantity of water is added to the mixture to make the ratio of milk and water in the new mixture 1:2.
Since the resulting amount of water must be double the original 2 liters of milk, the resulting ...”

“Let the 7 cars be A, B, C, D, P, Q and S.
Since P and Q must occupy adjacent positions, consider PQ a single element in the arrangement.
The number of ways to arrange the 6 elements A, B, C, D, PQ and S = 6! = 720.
In 1/2 of these arrangements, S will be to the LEFT of PQ.
In the ...”

“Let:
W = the total number of white marbles.
E = the total number of even marbles.
WE = the total number of white even marbles.
Statement 1:
Thus, WE = 0.
No way to determine the value of W or E.
INSUFFICIENT.
Statement 2:
Implication:
W-E = (0.2)(25) = 5.
Case 1: W=10, E=5, WE=0
...”

“Statement 2: Brittany always leaves the hotel at exactly 5.30pm.
To see the situation more clearly, plug in a value for the distance between the hotel and the office.
Let the distance = 6 miles.
Since Allison takes 1 hour -- from 5 to 6pm -- to travel the entire distance, Alison''s rate = d/t = ...”

“It is possible for back-to-back modifiers joined solely by a comma to have the same referent.
Three examples:
http://www.beatthegmat.com/the-hognose-snake-puts-on-an-impressive-bluff-t93558.html
http://www.beatthegmat.com/parallelism-issue-t96011.html
...”

“The usage of the term arrangements is misleading. The prompt seems intended to ask the following:
The first player selected can be paired with any of the 5 other players.
Thus, the number of options for the first player selected = 5.
At this point, 4 players remain.
The next player ...”

“The determine the percent decrease, we need to know the RATIO between the remaining amount of sugar after cookie-making and the original amount of sugar.
Statement 1:
Case 1: Remaining sugar after cookie-making = 10 pounds
Since the remaimning amount of sugar would need to increase by 30% to ...”

“Statement 1: The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend.
Case 1: Distance between Townsend and Smallville = 80 miles
Time to travel from Townsend to Smallville at a speed of 40 mph = 80/40 = 2 hours.
Time to travel from Smallville to Townsend = ...”

“Let x = the number of 100-point labs and y = the number of 85-point labs.
Question stem, rephrased:
What is the value of x?
Statement 1:
100x + 85y = 1140
20x + 17y = 228
The value in blue implies the following:
Since 20x will have a units digit of 0, 17y must have a UNITS DIGIT OF 8.
...”

“Alternate approach:
Since an absolute value cannot be negative -- implying that the right side of the equation must be NONNEGATIVE -- we know that x≥0.
If x²-6=x, we get:
x²-x-6=0
(x-3)(x+2) = 0
Only the factor in green implies a nonnegative option for x:
x-3=0
x=3.
The correct ...”

“The usage of number implies that we should consider NON-INTEGER values.
Both statements are satisfied by z=18.1 and z=19.1
Since the units digit of z can be different values, the two statements combinecd are INSUFFICIENT.
The correct answer is E.”

“Any BAG taken on the subway IS subject to search by the police.
Any BAGS taken on the subway ARE subject to search by the police.
In each of these sentences, any is a MODIFIER describing the subject in blue.
In the first sentence, the verb (is) is SINGULAR to match the singular subject in blue.
...”

“Statement 1:
No information about Charlie or Daryl.
INSUFFICIENT.
Statement 2:
For every 2 home run hits made by Charlie and Daryl combined, Brady made 4 home run hits, implying that Brady made 4 of every 6 home run hits:
4/6 = 2/3.
SUFFICIENT.
The correct answer is B.”

“Downhill speed - uphill speed = (x+10) - (x-8) = 18.
Implication:
The downhill speed is 18 mps greater than the uphill speed.
We can PLUG IN THE ANSWERS, which represent the downhill speed.
Since the downhill speed is 18 mps greater than the uphill speed, the correct answer must be GREATER ...”

“Statement 1:
To MINIMIZE the number of coins so that a sum of 170 cents is yielded by at most 8 coins, we must MAXIMIZE the number of quarters.
Since 7*25 = 175 cents, the number of quarters must be less than 7.
170 cents will be yielded by 6 quarters (totaling 150 cents) and 2 dimes (totaling 20 ...”

“Let the tank = 36 gallons.
Since the first pipe takes 3 hours to fill 1/2 the tank -- and (1/2)(36) = 18 gallons -- the rate for the first pipe = w/t =18/3 = 6 gallons per hour.
Since the second pipe takes 6 hours to fill 2/3 of the tank -- and (2/3)(36) = 24 gallons -- the rate for the second ...”

“Strategy:
Try to show that four of the answer choices do NOT have to be positive.
Case 1: x=1 and y=1
In this case, A, B and D are not positive.
Eliminate A, B and D.
Case 2: x=1 and y=2
In this case, C is not positive.
Eliminate C.
The correct answer is E.”

“Total amount of feed = (number of chickens)(number of days).
The number of chickens is INVERSELY PROPORTIONAL to the number of days.
If the number of chickens DOUBLES, then the feed will last for 1/2 the number of days.
If the number of chickens TRIPLES, then the feed will last for 1/3 the ...”

“Let J = the number of hours that John works.
Since John assembles 3 chairs per hour, the number of chairs assembled by John = 3J.
Let L = the number of hours that Larry works.
Since Larry assembles 4 chairs per hour, the number of chairs assembled by Larry = 4L.
Since John and Larry work a total ...”

“Number of options for Phoebe = 1. (Must be Margherita.)
Number of options for Ross = 3. (Of the 4 remaining pizzas, any but Hawaiian.)
Number of options for Chandler = 3. (Any of the 3 remaining pizzas.)
Number of options for Joey = 2. (Either of the 2 remaining pizzas.)
Number of options for ...”

“One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time.
Since replacing 1 man with 2 girls does not change the amount of time required to complete the job, 1 man is the equivalent of 2 ...”

“This is a MIXTURE problem.
Two speeds (3 mph and 8 mph) are combined to form a mixture with an average speed of 4 mph.
To determine how much time must be spent at each speed, we can use ALLIGATION.
Let U = the uphill speed and D = the downhill speed.
Step 1: Plot the 3 speeds on a number ...”

“2 and 17 have no factors in common.
Since n must be divisible by both 2 and 17 -- and 2*17 = 34 -- n must be a MULTIPLE OF 34.
Thus, n must be divisible by I.
Eliminate any answer choice that does not include I.
Eliminate B and E.
If n=34, then n is not divisible by II or III.
Eliminate any ...”

“To keep track of the number of options for each position, draw a TREE.
Start with the MOST RESTRICTED position, which is SWEEPER.
If Sweeper = 3, then Goalie = 5.
If Sweeper = 4, then Fullback = 2.
Here''s the tree so far:
http://s11.postimage.org/laxnlsbof/soccer_half_tree.jpg
Now complete ...”

“A group...EACH has a team.
The wording above does not convey a logical meaning.
When each serves as an adverb, it cannot refer to a singular noun such as a group.
When each serves as an adverb, it may refer only to a PLURAL noun.
The prompt seems intended to convey the following:
Since ...”

“Statement 1:
For every $100 commission that Dan earns, Jeff earns $120 in commission.
Since the commission rates for Dan and Jeff are unknown, no way to determine whether Jeff recorded more sales revenue than Dan.
INSUFFICIENT.
Statement 2:
Since Jeff''s sales and Dan''s sales can be any ...”

“Standard deviation describes how much the values in a set deviate from the mean. A larger standard deviation indicates that the values are deviating more -- getting farther away from -- the mean. So the question can be rephrased:
Do the values in set A deviate more from the mean than the do ...”

“x² < y² can be rephrased as follows:
|x| < |y|
Thus, the question stem can be rephrased as follows:
Is |x+10| < |y-10|?
|a-b| = the distance between a and b
|a+b| = |a-(-b)| = the distance between a and -b
Thus, the question stem can be further rephrased as follows:
Is the ...”

“The team has won 58 percent of the games it has played.
58% = 58/100 = 29/50.
The fraction in blue implies that the number of wins must be a MULTIPLE OF 29:
If 50 games are played, the number of wins = (29/50)(50) = 29.
If 100 games are played, the number of wins = (29/50)(100) = 58.
If 150 ...”

“The SC above is modeled -- not altogether successfully -- after SC30 in the OG12:
https://www.beatthegmat.com/not-sure-about-the-answer-of-a-question-from-og12-t265701.html
Rather than study the SC above, stick to the official SC.”

“The values in the chart DOUBLE-COUNT any student who belongs to 2 of the 3 clubs.
The totals for chess and drama -- 40 and 30 -- double-count the 10 students who belong to both chess AND drama.
The totals for chess and math -- 40 and 25 -- double-count the 5 students who belong to both chess AND ...”

“Let r=9, with the result that 1+r = 10.
Her husband can turn each dollar she gives him into 1+r, which will be deposited in a bank account.
Let the wife give the husband $1, implying that the husband turns this $1 into 1+r = $10.
Thus, the amount deposited into the account = $10.
The ...”

“Let the distance between A and B = 120 miles.
Since the 120-mile distance between A and B is twice the distance between B and C, the distance between B and C = 60 miles, implying that the total distance = 120+60 = 180 miles.
Since the average speed for the 120-mile leg between A and B = 60 ...”

“The sum for Lisa and Mike is 50 less than the sum for Norm and Mike.
Since Mike is common to both sums, the 50-photo difference must be yielded solely by Lisa and Norm.
Implication:
Lisa took 50 fewer photos than Norm.
We can PLUG IN THE ANSWERS, which represent the number of photos taken ...”

“Let h=3.
Since an average of $78 is sold per hour, the total revenue for 3 hours = 3*78 = 234.
Since the cost for the day is composed of $150 for the table rental and $10 per hour for the assistant, the cost for the day = 150 + 3*10 = 180.
P = profit = revenue - cost = 234 - 180 = 54.
The ...”

“It takes for granted that X implies that X is an ASSUMPTION: a statement that must be true for the conclusion to hold.
Thus, when X is negated, the conclusion must be invalidated.
B: It takes for granted that the volunteers with the highest levels of self-esteem had better social skills than did ...”

“Premise:
Volunteers with the highest levels of self-esteem consistently rated themselves as having much better social skills than did the volunteers with moderate levels.
Conclusion:
Attaining an exceptionally high level of self-esteem greatly improves one’s social skills.
The correct answer ...”

“Let k=10.
2k years ago Frank was 3k years old.
Since k=10, 20 years ago Frank was 30 years old, implying that his age now = 30+20 = 50.
In k years Frank''s age, in years, will be:
Since Frank is now 50 years old, Frank''s age in 10 years = 50+10 = 60.
The correct answer must yield a value ...”

“Alternate approach:
For any quadratic in the form x² + bx + c = 0, where b and c are rational numbers:
If one of the roots is m+√n, then the other root is m-√n.
Sum of the roots = -b
Product of the roots = c
Here, since one of the roots is 1+√2, the other root is 1-√2.
Thus:
-b ...”

“Since each payday is $10 more than the preceding payday, the list of 15 paydays constitutes an EVENLY SPACED SET.
Statement 1:
For any evenly spaced set:
Median = Sum/Quantity.
Thus:
Pay on the 14th day = median of the 3 most recent days = sum/quantity = 690/3 = 330.
Since the pay on the ...”

“Original deposit: 10,000.
The balance doubles every 10 years.
After 10 years, we get:
2*10,000 = 20,000.
After 20 years, we get:
2*20,000 = 40,000.
After 30 years, we get:
2*40,000 = 80,000.
The correct answer is E.”

“Let the 6 vertices = A, B, C, D, E and F.
P(the triangle includes AT LEAST 1 side of the hexagon) = 1 - P(the triangle includes NO sides of the hexagon).
From the 6 vertices, the number of ways to choose 3 to form a triangle = 6C3 = (6*5*4)/(3*2*1) = 20.
Of these 20 triangles, only 2 include ...”

“Number of options for the 1st person = 8.
Number of options for the 2nd person = 6. (Of the 7 people left, we can''t use the twin of the 1st person chosen, leaving 7-1= 6 choices.)
Number of options for the 3rd person = 4. (Of the 6 people left, we can''t use the twins of the 2 people already ...”

“Let x = the yearly increase.
Height after 4 years = 4 + 4x.
Height after 6 years = 4 + 6x.
At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year.
The phrase in red implies the following:
The height in the 6th year was 6/5 of the height in the 4th ...”

“Any integer that is a multiple 10 is also a multiple of 5.
Implication:
Any integer that satisfies Statement 1 will also satisfy Statement 2.
Both statements are satisfied by the following cases:
Case 1: p=10, which is a multiple of 10 (and thus is also a multiple of 5)
In this case, p is NOT a ...”

“The drawings below satisfy both statements:
http://s4.postimage.org/1yjxdhskk/ABCD_number_line.jpg
Since in the first case BD=1 and in the second case BD=17, the two statements combined are insufficient.
The correct answer is E.”

“Case 1: Team A wins a prize, with the result that Team B must win a prize
Number of prizes that could be won by A = 3.
Number of remaining prizes that could be won by B = 2.
Number of teams that could win the third prize = 4. (Any of the 4 remaining teams.)
To combine these options, we ...”

“Let n=50, implying that 50% of the sold juice packs are A, while the other half are B.
In other words, the store sells an EQUAL NUMBER of each type of juice pack.
Let the store sell 2 OF EACH TYPE of juice pack.
Revenue from two $1 packs of A = 2*1 = 2.
Revenue from two $1.5 packs of B = ...”

“Let James'' rate = 3 mph.
James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James stated from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James ...”

“The equation for a circle with its center at the origin and a radius of r is as follows:
x² + y² = r².
Thus, x² + y² = 25 is a circle with its center at the origin and a radius of 5.
A point on this circle will have integer coordinates in three cases:
Case 1: The radius is horizontal
...”

“The average speed -- 2.8 miles per hour -- must be BETWEEN the two individual rates (s and s+1).
Thus, s = 2 miles per hour and s+1 = 3 miles per hour.
We can PLUG IN THE ANSWERS, which represent the fraction traveled at 2 miles per hour.
When the correct fraction is plugged in, the average ...”

“Number of options for president = 10. (Any of the 10 members.)
From the remaining 9 members, the number of ways to choose 2 to serve as vice-president = 9C2 = (9*8)/(2*1) = 36.
To combine these options, we multiply:
10*36 = 360.
The correct answer is E.”

“One approach is to use the following equation:
(machines)(time) / output = (machines)(time) / output
In the equation above:
Machines and time are INVERSELY PROPORTIONAL.
As the number of machines increases, the amount of time required to produce the same output decreases.
Machines and ...”

“One approach:
Good Cases = Total Cases - Bad Cases.
Total Cases:
From 9 points, the number of ways to choose 3 = 9C3 = (9*8*7)/(3*2*1) = 84.
Bad Case 1: Choosing 3 collinear points from the 5-point line, with the result that a triangle cannot be formed
From the 5 points on this line, the ...”

“Since the prompt indicates that n is an integer greater than 1, the case in red is not possible.
The only value that satisfies Statement 2 is n=2, making Statement 2 SUFFICIENT.
The correct answer thus is not C but B.”

“Between 1982 and 1987 are 4 non-leap years and 1 leap year.
Each non-leap year is composed of 365 days; a leap year is composed of 366 days.
Thus, from June 25, 1982 to June 25, 1987, the total number of days = 4(365) + 366.
Since June 25, 1982 is a Friday, every passing of 7 days will yield ...”

“(man''s rate)/(woman''s rate) = (4 mph)/(20 mph) = 1/5.
Between the moment when the woman passes the man and the moment when she stops to wait, the woman and the man each travel for 5 minutes.
During these 5 minutes, the woman travels x miles.
Since the man''s rate is 1/5 the woman''s rate, the ...”

“Statement 1:
Here, n can be any prime number.
INSUFFICIENT.
Statement 2:
(n+2)/n = n/n + 2/n = 1 + 2/n.
Given that n must be an integer, 1 + 2/n will yield an integer value only if n is a FACTOR OF 2.
Factors of 2: 1 and 2.
Since n must be greater than 1, n=2.
SUFFICIENT.
The correct ...”

“Statement 1:
Rate and time have a RECIPROCAL relationship.
A rate ratio of (40kph) : (50kph) = 4:5 implies a time ratio of 5:4 = 10:8.
The time ratio in blue implies that -- for the time at 40kph to be 2 hours more than the time at 50kph -- the time at 40kph must be 10 hours, while the time at ...”

“Statement 1:
The TIME RATIO for Tim and Glenn = 48:60 = 4:5.
Since time and rate have a RECIPROCAL relationship, the RATE RATIO for Tim and Glenn = 5:4.
Implication:
For every 5 laps that Tim runs, Glenn runs only 4 laps.
Thus, after running 5 laps, Tim will have run 1 more lap than Glenn. ...”

“Let the one-bedroom rent = $2 per month, implying that the two-bedroom rent = 2 + (50% of 2) = 2 + 1 = $3 per month.
Since the rent for a one-bedroom apartment = $2 per month, the total amount to rent a one-bedroom apartment for 12 months = 2*12 = $24.
For $24, the number of months yielded for a ...”

“Let W = Winesap crates, M = McIntosh crates, and R = Roma crates.
Since a total of 25 crates are ordered, we get:
W + M + R = 25.
To MINIMIZE the value of W, we must MAXIMIZE the values of M and R.
Since the store is to order more crates of Winesap than crates of McIntosh and more crates of ...”

“The prompt states:
Every day, the two machines operate constantly from 08:00 to 18:00.
As shown in my post above, Statement 2 yields the following rate ratio:
(W alone) : (W and D together) = 1:5.
The rate ratio indicates that -- of every 5 kg produced when W and D work together from 8:00 to ...”

“Generally, for + VERBing is an ADJECTIVE serving to modify a NOUN.
A: The Breeding Bird Survey uses annual roadside counts along established routes for monitoring.
Here, for monitoring seems to be an ADJECTIVE modifying routes, implying that the ROUTES were designed FOR MONITORING.
Not the ...”

“Let x = the two-digit number.
Since dividing by 4 leaves a remainder of 1, x is equal to one more than a multiple of 4:
x = 4k + 1
Since dividing by 14 leaves a remainder of 1, x is equal to one more than a multiple of 14:
x = 14m + 1
Since the two expressions in blue must be equal, we ...”

“Statement 1: If the selling price per coat had been twice as much , the store’s gross profit on the 20 coats would have been $2,400
When the selling price per coat is doubled, the resulting profit per coat = (total profit)/(number of coats) = 2400/20 = 120.
Case 1: Double the selling price = ...”

“60! = 60*59*58*....*3*2*1.
Since 10=2*5, EVERY COMBINATION OF 2*5 contained within the prime-factorization of 60! will yield a 0 at the end of the integer representation of 60!.
The prime-factorization of 60! is composed of FAR MORE 2''S than 5''s.
Thus, the number of 0''s depends on the ...”

“An INSCRIBED ANGLE is formed by two chords.
A CENTRAL ANGLE is formed by two radii.
When an inscribed angle and a central angle intercept the same two points on a circle, the inscribed angle is 1/2 the central angle.
https://i.postimg.cc/qgbxWKdW/Angle-CEB-1.png
Here, inscribed angle CFB and ...”

“Statement 1: x=12u, where u is an integer and x=8y+12.
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.
If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.
If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.
Since the ...”

“Time and rate have a RECIPROCAL RELATIONSHIP:
2 times as fast implies 1/2 the time.
3 times as fast implies 1/3 the time.
Let t = the normal time.
Since the bus travels today at 7/6 its normal rate, the time today is equal to 6/7 of the normal time:
(6/7)t.
Since the bus leaves 1/2 hour late ...”

“Given any set that is symmetrical about the median:
Sum = (quantity)(median)
The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222):
...212, 213, 221, 222, 223, 231, 232...
Thus:
Sum of the 27 integers = (quantity)(median) = 27*222 = ...”

“For the purposes of a Y/N DS problem, the rephrase is the equivalent of the original question stem.
Original question stem: Is x² > xy?
The answer will be YES if y/x is less than 1..
The answer will be NO if y/x is NOT less than 1.
Thus, the original question stem can be rephrased as ...”

“x² > xy is true only if x is NONZERO, in which case we can safely divide by x², since the square of a nonzero value must be positive:
x²/x² > (xy)/x²
1 > y/x
Question stem, rephrased:
Is y/x < 1?
Statement 1: x > y
If x=2 and y=1, then y/x =1/2. so the answer to the ...”

“Let the job = the LCM of the 3 times = 60 widgets.
Maximum time:
The maximum time is yielded when the slowest person -- R -- works alone.
Since R takes 15 hours to complete the 60-widget job, the maximum time = 15 hours.
Minimum time:
The minimum time is yielded when all 3 people work ...”

“X led to Y implies that X did not directly cause Y but that X was a factor that helped to yield Y.
SC755 in the OG17:
An inability to adapt led to their extinction.
Here, an inability to adapt did not directly cause the extinction but was a factor that helped to yield the extinction.
A cause ...”

“Generally, COMMA + as-clause implies that the clause after as LEADS to the preceding clause, with the result that the two clauses happen concurrently.
An OA in GMATPrep:
The steel industry has changed radically over the last two decades, as large, integrated companies such as Bethlehem Steel have ...”

“Let B = Barry''s rate and W = the walkway''s rate.
The distance can be ANY VALUE.
Let the distance = 240 meters.
WITH the walkway, the time = 30 seconds:
Here, Barry and the walkway WORK TOGETHER, so we ADD their rates:
B+W = d/t = 240/30 = 8 meters per second.
AGAINST the walkway, the ...”

“Let the x family members = the product of the two denominators = 3*7 = 21.
Number who chose to have a hamburger = (1/3)(21) = 7.
Since 1/7 of the hamburger-eaters also had a hot dog, the number who had both = (1/7)(7) = 1.
Thus, the number who did NOT have both = (total members) - (number who had ...”

“I believe that the prompt above is outdated; regardless, it does not list a correct answer. The prompt below appears in Manhattan''s 5 lb. Book of GRE Practice Problems:
Case 1: 57XXX
Number of options for the hundreds digit = 10. (Any digit 0-9.)
Number of options for the tens digit = 10. ...”

“Since the N cookies can be divided evenly among 24 or 18 students, N must be divisible by both 24 and 18.
One way to determine the LCM of two integers is to take multiples of the LARGER integer until we get a multiple of the SMALLER integer.
Multiples of 24:
24, 48, 72...
We can stop here: the ...”

“Statement 1:
If NONE of the members sells at least 2 tickets, then the 3 members sell a maximum of 1 ticket each, yielding the following total:
1+1+1 = 3.
Too small, since the total number of tickets sold = 6.
Implication:
For a total of 6 tickets to be sold, one of the 3 members must sell at ...”

“This is a weighted average/mixture problem.
Average score for the first N games = 35.
Score for the final game = 28.
Average score for the MIXTURE of all the games = 34.
Step-by-step instruction for the alligation approach:
Step 1: Plot the 3 averages on a number line, with the average ...”

“The difference between the endpoints of the range = 200-10 = 190.
Since neither endpoint is a multiple of 3, we can count the multiples of 3 simply by dividing 3 into 190:
190/3 ≈ 63.
The correct answer is D.”

“Let d=100.
After the value of the stock increases by 40% to $140, the share received by each of the 20 friends = 140/20 = 7.
The correct answer must yield a value of 7 when d=100.
Only E works:
(7d)/100 = (7*100)/100 = 7.
The correct answer is E.”

“We can PLUG IN THE ANSWERS and use alligation.
To learn about alligation, check my posts here:
https://www.beatthegmat.com/statistics-average-question-t288903.html
https://www.beatthegmat.com/alligation-method-t293459.html
https://www.beatthegmat.com/mean-t266852.html
Let O = the original ...”

“Statement 1:
Time and rate have a RECIPROCAL relationship.
Since X and Y together take 2/3 as long as Y alone, X and Y together work 3/2 as fast as Y alone.
Thus, if Y''s rate alone = 2 units per hour, then X and Y''s combined rate = (3/2)(2) = 3 units per hour, implying that X''s rate alone = ...”

“To form a triangle, we must select 3 points such that at most 2 are collinear.
Case 1: Select 3 points not on the x-axis
From the 4 points not on the x-axis, the number of ways to choose 3 = 4C3 = (4*3*2)/(3*2*1) = 4.
Case 2: Select 1 point on the x-axis and two points not on the x-axis
From ...”

“Number of options for president = 8. (Any of the 8 candidates.)
Number of options for treasurer = 7. (Any of the 7 remaining candidates.)
From the 6 remaining people, the number of ways to choose 3 to serve as warrant officers = 6C3 = (6*5*4)/(3*2*1) = 20.
To combine these options, we multiply:
...”

“For the set of pencils to be shared evenly, the total number of pencils must be divisible by the total number of children.
Thus, the correct answer must be divisible by 2, 3, 4, 5 and 6.
Of the five answer choices, only E is divisible by 2, 3, 4, 5 and 6.
The correct answer is E.”

“We can PLUG IN THE ANSWERS, which represent the length of the shorter leg.
When the correct answer is plugged in, the area = 400.
B: 20
Longer leg = two more than three times the shorter leg = 2 + 3*20 = 62.
Area = (1/2)(20)(62) = 620.
Since the area is too great, a smaller answer choice is ...”

“Statement 1:
Test one case that also satisfies Statement 2.
Case 1: The frame is 8 inches tall, with the result that the total area of the frame and picture combined = 6*8 = 48
Area of the picture = (total area) - (shaded region) = 48-24 = 24.
Implication:
The width of the frame is 1 inch, ...”

“A general rule:
Given x^(even power) - y^(even power), where x and y are distinct positive integers, we can always simplify by applying the difference of two squares.
Each time we apply the difference of two squares to the expression in blue, the exponents are reduced by 1/2.
For example:
x⁸ - ...”

“For any evenly spaced set:
sum = (count)(median)
Sum of the first n even numbers = n * median of the first n even numbers
Sum of the last n odd numbers = n * median of the last n odd numbers
Since the ratio of the two sums = 16/85, we get:
(n * median of the first n even numbers) / (n * ...”

“Let Rex''s income = 100.
Since Quentin''s income is 60% less than Rex''s, Quentin''s income = 100 - (60% of 100) = 100-60 = 40.
Since Sam''s income is 25% less than Quentin''s, Sam''s income = 40 - (25% of 40) = 40-10 = 30.
After Rex gives 60% of his income to Sam, Sam''s new income = 30 + (60% ...”

“We can PLUG IN THE ANSWERS, which represent the value of x.
Since an absolute value cannot be equal to a negative value, the correct answer cannot be A or B.
D: x=2
If we plug x=2 into |x-6|=2x, we get:
|2-6| = 2*2
|-4| = 4
4 = 4.
Success!
The correct answer is D.”

“Let the distance between A and B = 120 miles.
Since the 120-mile distance between A and B is twice the distance between B and C, the distance between B and C = 60 miles, implying that the total distance = 120+60 = 180 miles.
Since the average speed for the 120-mile leg between A and B = 60 mph, ...”

“The answer choices are already phrased as statements.
The OA is a statement that weakens the conclusion.
OA: Computer owners in the United States predominately belong to a demographic group that have long been known to spend less time watching television.
This statement indicates that a lack of ...”

“Let the distance between A and B = the LCM of the two speeds = 5*6 = 30 kilometers, implying that the distance for the entire round trip = 60 kilometers.
At a rate of 5 kph, the time to travel the 30 kilometers from A to B = d/r = 30/5 = 6 hours.
At a rate of 6 kph, the time to travel the entire ...”

“In ascending order of EMPLOYEES PER PROJECT, let the projects be numbered 1 through 100:
1, 2, 3.....49, 50, 51, ....98, 99, 100.
Projects 50 and 51 determine the MEDIAN number of employees:
Question rephrased: How many employees are assigned to PROJECTS 50 and 51?
Looking at the two ...”

“The sum for Lisa and Mike is 50 less than the sum for Norm and Mike.
Since Mike is common to both sums, the 50-photo difference must be yielded solely by Lisa and Norm.
Implication:
Lisa took 50 fewer photos than Norm.
We can PLUG IN THE ANSWERS, which represent the number of photos taken by ...”

“Circle ABCD:
The equation for a circle with a center at the origin and a radius of r is x² + y² = r².
The equation for circle ABCD is x² + y² = 5².
Thus, circle ABCD has a radius of 5.
Line EF:
3y = 4x + 25
y = (4/3)x + 25/3
Slope = 4/3.
A radius drawn to a point of tangency forms ...”

“In many cases, similar or identical solutions are posted when two experts respond at virtually the same time, with the result that the second poster is not aware of the first.
The time stamps for the two solutions here indicate that both solutions were posted at 11:45am.”

“The wording in Statement 1 does not convey a clear meaning:
For every value in the list, if the value is increased by 1, the number of distinct values in the list does not change.
Here, the term in red could convey to some test-takers that EVERY VALUE in the list is to be increased by 1.
Not the ...”

“Statement 1:
If profits increase by $X from the first year to the second year, then they must increase by $0 from the second year to the third year, with the result that the increase in profits from the first year to the third year (X+0 = $X) is the same as the increase in profits from the first ...”

“Statement 1:
25Q + 10D = 170
5Q + 2D = 34
To minimize the number of coins, we must maximize the value of Q.
If Q=6, then D=2, with the result that 5Q + 2D = (5*6) + (2*2) = 34.
In this case, the total number of coins = 6+2 = 8.
If the value of Q decreases, then the total number of coins will ...”

“having + VERBed is a viable construction.
An introductory HAVING + VERBed modifier implies PRIOR ACTION: an action completed before the main verb.
HAVING FINISHED his dessert, John LEFT the table.
Here, HAVING FINISHED implies PRIOR ACTION -- an action completed before John LEFT the table.
...”

“Since 6% of the 1000-gallon tank is fluorine, the amount of fluorine = (6/100)(1000) = 60 gallons.
Since the total volume decreases at a rate of 10 gallons per day for 25 days, the decrease in the total volume = 10*25 = 250 gallons.
Remaining total volume = 1000-250 = 750 gallons.
Since 60 of the ...”

“CRITICAL POINTS in an equality occur when the two sides are EQUAL or the inequality is undefined.
Here, the two statements provide the critical points for the inequality in the prompt:
When x=0, the two sides are equal.
When x=1, the inequality is undefined.
To determine the range(s) where the ...”

“Statement 1:
For every $100 commission that Dan earns, Jeff earns $120 in commission.
Since the commission rates for Dan and Jeff are unknown, no way to determine whether Jeff recorded more sales revenue than Dan.
INSUFFICIENT.
Statement 2:
Since Jeff''s sales and Dan''s sales can be any ...”

“Let a=b=0.5, implying that the rectangle''s length = a+b = 0.5+0.5 = 1 and that its area = (1/0.5) + (1/0.5) = (10/5) + (10/5) = 2+2 = 4.
Since the rectangle has an area of 4 and a length of 1, its width = 4.
The correct answer must yield a value of 4 when a=b=0.5.
Only E works:
1/ab = 1/(0.5 * ...”

“N/J = 134.08 = 13408/100 = 3352/25.
The resulting ratio indicates that N must be multiple of 3352 and that J must be a multiple of 25.
Of the five answer choices, only E is a multiple of 25.
The correct answer is E.”

“We can PLUG IN THE ANSWERS, which represent the number of blue marbles in the second container.
When the correct answer is plugged in, the probability of selecting 2 red marbles OR 2 blue marbles = 1/2.
B: 4 blue marbles, implying a total of 10 marbles in the second container
Case1: 2 red ...”

“=
We can PLUG IN THE ANSWERS, which represent the number of tulips.
When the correct answer is plugged in, the total cost = $48.
D: 24 tulips, implying 16 roses, for a total of 40 flowers
Since each of the 24 tulips costs $1, and each of the 16 roses costs $1.50, we get:
Total cost = (24*1) ...”

“Let r=100 cents.
For 3 people, the cost per mile = 3*100 = 300 cents = 3 dollars.
Let x = 2 miles.
Regular cost for 3 people to travel 2 miles = 3*2 = 6 dollars.
With a 50% discount, the cost in dollars = 3. This is the target.
Plug x=2 and r=100 into the answers to see which yields the ...”

“We can PLUG IN THE ANSWERS, which represent the length.
When the correct answer is plugged in, perimeter = 360.
B: 140
Since the length is twice the width, W = 70, with the result that the perimeter = 140+140+70+70 = 420.
Since the perimeter is too great, a SMALLER answer choice is needed.
...”

“Generally, a COMMA + VERBing modifier serves to modify the nearest preceding action and the agent of this action.
C: instructions that...will help merchants confirm a check''s authenticity, by including reminders to watch the endorsement.
.Here, COMMA + by including seems to modify merchants ...”

“When a positive integer is divided by 100, the remainder is yielded by the last two digits:
123/100 = 1 R23
548/100 = 5 R48
692/100 = 6 R92
Thus:
The remainder when 7¹⁰ is divided by 100 is equal to the last two digits of 7¹⁰.
Calculate the last two digits for consecutive powers of 7 ...”

“TIME and RATE have a reciprocal relationship.
Statement 1:
The time ratio for Barbara and Alice = (14 hours)/(21 hours).
The rate ratio for Barbara and Alice is equal to the reciprocal of the time ratio:
(B''s rate)/(A''s rate) = 21/14 = 3/2.
Let B = 3 units per hour and A = 2 units per ...”

“Let James'' rate = 3 mph.
James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James stated from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James ...”

“Each of the 10 members has an EQUAL chance of being elected secretary.
P(Harry is elected secretary) = 1/10.
Each of the 10 members has an EQUAL chance of being elected treasurer.
P(Harry is elected treasurer) = 1/10.
Since EITHER outcome is favorable, we ADD the fractions:
1/10 + 1/10 = 2/10 = ...”

“The conclusion is constrained to the case at hand:
Of the 10 babies recently placed with parents, 8 were given to friends of the staff.
The assumption is that the premise (the parents given babies -- including the 8 friends --all far surpassed the agency''s criteria) is sufficient to prove the ...”

“Let:
b = the number of boys
g = the number of girls
s = the number of cents for each soda
j = the number of cents for each juice box
Note:
All of the values above must be POSITIVE INTEGERS.
Case One: If every boy in a kindergarten class buys a soda and every girl in the same class buys a ...”

“Let X = the number of 52¢ lemonades and Y = the number of 58¢ lemonades.
Since the question stem asks for the number of 52¢ lemonades, we get:
What is the value of X?
Statement 1:
X+Y = 9
Since X can be any value between 0 and 9, inclusive, INSUFFICIENT.
Statement 2:
52X + 58Y = 492
...”

“Alternate approach:
The 3 red cars must occupy a combination of 3 spaces.
From 5 spaces, the number of ways to choose a combination of 3 for the red cars = 5C3 = (5*4*3)/(3*2*1) = 10.
Number of options for the blue car = 2. (Either of the 2 remaining spaces.)
Number of options for the yellow ...”

“Premise:
All of the applicants given babies far surpassed the adoption criteria.
Conclusion:
In placing 8 of the 10 babies with friends, the staff was not guilty of favoritism.
Apply the NEGATION TEST.
When the correct answer is negated, the conclusion will be invalidated.
B, negated:
Of ...”

“In other words:
How many ways can 5 girls and 5 guys be divided into pairs, if each pair must consist of one girl and one guy?
Number of options for the first girl = 5. (Any of the 5 guys.)
Number of options for the second girl = 4. (Any of the 4 remaining guys).
Number of options for the ...”

“The average of s and 10 lies HALFWAY between s and 10 on the number line:
s------------AVERAGE------------10
Statement 1:
Since t closer to 10 than to s, t must be greater than the average of s and 10, as shown here:
s-----------AVERAGE---t----------10
Thus, the answer to the question stem is ...”

“Good committees = total committees - bad committees.
Total committees:
From 10 people, the number of ways to choose 4 = 10C4 = (10*9*8*7)/(4*3*2*1) = 210.
Bad committees:
Since a good committee includes at most 1 married couple, a bad committee is composed of 2 married couples.
From 3 ...”

“For any evenly spaced set:
sum = (number)(median)
We can PLUG IN THE ANSWERS, which represent the sum.
Since the number of integers can be 5 or 7, the sum must be a multiple of both 5 and 7.
Eliminate A, which is not divisible by 5.
Eliminate D, which is not divisible by 7.
The equation in ...”

“THE INSURED is the person covered by the insurance.
Plan A:
If $1000 is LESS than 50% of the cost level, then THE INSURED pays $1000, and PLAN A pays the remainder.
If $1000 is MORE than 50% of the cost level, then THE INSURED pays 50% of the cost level, and PLAN A pays the remainder.
Plan ...”

“Statement 1 indicates that 80 grams of water are added to the 80 grams of original solution, DOUBLING the total volume from 80 grams to 160 grams
Statement 2 indicates that the added water reduces the acid concentration by 1/2.
Doubling the total volume will reduce the acid concentration by 1/2 ...”

“We can PLUG IN THE ANSWERS, which represent the original number of trees.
Since the number of trees increases by 1/4 each year, the correct answer must be a multiple of 4.
The last 2 digits of a multiple of 4 must themselves form a multiple of 4.
Eliminate A (1250) and C (2250), since 50 is ...”

“Case 1:
Every 6-block rectangle with a base of 2 and a height of 3 yields 4 possible L-shapes:
https://i.postimg.cc/1gJCfb5J/chessboard_L_shape_1.png
Chessboard:
https://i.postimg.cc/LYJThXW8/chessboard_whole.png
The base for the 6-block rectangle shown above can appear in rows 1 through 6, ...”

“During each of Amit''s 4 shifts, he paints 1/2 of the room, leaving 1/2 the room unpainted.
Since 1/2 of the room is left unpainted FOUR TIMES, the total fraction left unpainted as a result of Amit''s 4 shifts = (1/2)⁴.
Since Ian works after each of Amit''s first 3 shifts, Ian paints a total ...”

“Since the answer choices are in terms of a variable -- the value of r -- we can PLUG IN.
Let the ENTIRE WIRE be used to form the square.
Then:
Perimeter of the square = 40.
Side = 10.
Area = 100.
Circle area + square area = 0 + 100 = 100. This is our target.
Since the circle has no ...”

“x²y² will be even unless both x and y are ODD.
Question stem, rephrased:
Are x and y both odd?
Statement 1:
x+5 is prime number greater than 2 and thus must be ODD.
Implication:
x+5 = ODD
x = ODD - 5 = ODD - ODD = EVEN.
Since x is even, x and y cannot both be odd.
SUFFICIENT.
...”

“Rate and time have a RECIPROCAL RELATIONSHIP.
The RATE RATIO for the hiker and cyclist = 4 mph : 20 mph = 1:5.
Thus:
The TIME RATIO for the hiker and cyclist = the reciprocal of the rate ratio = 5:1.
Implication:
The hiker will take 5 TIMES AS LONG as the cyclist to travel the same ...”

“Alcohol percentage in the original solution: 3%.
Alcohol percentage in the added water: 0%.
Alcohol percentage in the mixture: 1%.
Let S = the original solution and W = the added water.
The following approach is called ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.
Step 1: ...”

“When the same distance is traveled at two different speeds, the average speed for the entire trip will be a bit LESS than the average of the two speeds.
Since (70+50)/2 = 60, the average speed for Vivian''s''s entire trip must be a bit less than 60.
The correct answer is B.”

“An alternate approach is to PLUG IN THE ANSWERS, which represents Tom''s current age.
When the correct answer is plugged in, Mom''s age in 6 years will be 3 times Tom''s age in 6 years.
D: Tom = 6
Since Mom is currently 5 times as old as Tom, Mom = 5*6 = 30.
Tom''s age in 6 years = 6+6 = 12.
...”

“First 8 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19.
The sum will be even if 2 is among the three numbers selected, with the result that the sum = EVEN PRIME + ODD PRIME + ODD PRIME = EVEN.
Since 3 NUMBERS are selected from 8 OPTIONS, the probability that 2 will be among the three numbers selected = ...”

“Let T = the total number of employees.
Since 60% of the employees are over 40, we get:
Over 40 = 0.6T
30% of those over 40 -- in other words, 30% of the expression in blue -- have master''s degrees.
Thus:
Over 40 with master''s = 0.3(0.6T).
To calculate the value of the expression in ...”

“Since Pat travels at 30 miles per hour for the first 12 minutes -- the equivalent of 1/5 hour -- the distance traveled in the first 12 minutes = (30)(1/5) = 6 miles.
Since the remaining 14 miles are traveled at 8 miles per hour, the time to travel the remaining 14 miles = 14/8 = 7/4 hours = 1 hour ...”

“The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3.
Draw a VENN DIGRAM showing where the prime factors of n, 16 and 45 overlap:
https://i.postimg.cc/xkyNCFg7/GCF_Venn_n_16_and_45.jpg
The diagram implies the following:
Since n and ...”

“Good arrangements = total arrangements - bad arrangements.
Total arrangements:
Number of ways to arrange 5 elements = 5!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the ...”

“Let the total number of students = 100.
Since 50% of the students study science, the number of science students = (50/100)(100) = 50, implying that the number of non-science students = 100-50 = 50.
Since 30% of the 50 science students score more than 80%, the number of science students who ...”

“5-pound bags = ($13.85)/5 ≈ $3 per pound.
10-pound bags = ($20.43)/10 = more than $2 per pound.
25-pound bags = ($32.25)/25 = less than $1.50 per pound.
To minimize the cost, we must maximize the number of 25-pound bags, which offer seed at the lowest price per pound.
Two 25-pound bags = 50 ...”

“ALWAYS KEEP YOUR EYES ON THE ANSWER CHOICES.
C, D and E imply that the flower bed is probably a 6-8-10 triangle in which y=6, x = y+2 = 6+2 = 8, and z=10.
Check whether this triangle satisfies the condition that the flower bed has an area of 24 square yards:
Area of a 6-8-10 triangle = ...”

“Let the total number of chocolates = the LCM of the four denominators 2, 5, 6 and 12 = 60.
Number of chocolates given to the four children:
(1/2)(60) = 30
(1/5)(60) = 12
(1/6)(60) = 10
(1/12)(60) = 5
Total number of chocolates = 30+12+10+5 = 57.
The correct answer is C.”

“Of the units sold, non-condos constitute 85% = 85/100 = 17/20.
Implication:
Of every 20 units sold, 17 are non-condos, with the result that the number of non-condos must be a multiple of 17.
A. 16000
B. 13600
C. 12400
D. 11200
E. 8600
Only the portion in green is a multiple of 17:
...”

“Let G = at least 3 goals and NG = not at least 3 goals.
Let W = win the match and NW = not win the match.
Use a DOUBLE-MATRIX to organize the data:
https://i.postimg.cc/JHLj6Tyr/matrix_goals_wins_0.png
The probability that Ronaldo will score at least 3 goals in the match is 0.59.
The ...”

“Let S = the smallest number of L = the largest number.
Since the largest number is 4 more than 3 times the smallest number, we get:
L = 3S + 4
L-4 = 3S
(L-4)/3 = S.
Since the average is 50 -- and the numbers are not all the same -- S must be LESS THAN 50, while L must be greater than 50.
...”

“Once the child has been placed at the table:
Number of ways to arrange the 2 women in the 2 seats adjacent to the child = 2! = 2.
Number of ways to arrange the 4 men in the remaining 4 seats = 4! = 24.
To combine these options, we multiply:
2*24 = 48.
The correct answer is C.”

“Number of digit options for the repeated digit = 9. (Any digit but 0.)
Number of digit options for the non-repeated digit = 8. (Any digit but 0 and the digit already used.)
Number of position options for the non-repeated digit = 3. (The hundreds place, the tens place, or the units place.)
To ...”

“Algebra:
Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates.
(A+B)(3) = 1 widget.
If Machine A''s speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates.
(2A+B)(2) = 1 widget.
...”

“We can PLUG IN THE ANSWERS, which represent A''s time to produce 1 widget.
Since A and B working together take 3 hours to produce 1 widget, A working alone must take MORE than 3 hours.
Eliminate A, B and C.
Let the value of each widget = 30 units.
Since A and B together take 3 hours to produce ...”

“Use a DOUBLE-MATRIX to organize the data.
Statements combined:
Case 1: m=25% and c=25%
Total number of males = (25/100)(120) = 30.
Total number of custodial employees = (25/100)(120) = 30.
The following matrix is yielded:
https://i.postimg.cc/H88Lq5zN/matrix_custodial_1.png
In this case, ...”

“The default referent for a subject pronoun is the subject of the preceding INDEPENDENT CLAUSE.
A: Those who have public pensions are being pursued strongly by pension advance companies that operate without much oversight from banking regulators, but they are now drawing scrutiny.
Here, the default ...”

“This interpretation is not valid.
The comparative in blue is clearly an ADJECTIVE describing a city..
Since as big as refers not to China but to A CITY. no comparison between China and Singapore is implied.
The sentence unambiguously compares A CITY to Singapore, as discussed below.
...”

“Almost no math is needed here.
Answer choice A implies the following:
BCD is a 45-45-90 triangle in which each leg has a length of 1.
DEF is a 45-45-90 triangle in which each leg has a length of 5.
Answer choice E implies the following:
BCD is a 45-45-90 triangle in which each leg has a ...”

“In four years, Andy will be twice as old as Betsy.
Since Andy''s age in 4 years is twice Betsy''s age in 4 years, we get:
A+4 = 2(B+4)
A+4 = 2B+8
A-2B = 4.
Statement 1:
Since Andy''s age 4 years ago is twice Betsy''s current age, we get:
A-4 = 2B
A-2B = 4.
Same equation as given in the ...”

“Since 1000 - 101 ≈ 900 -- and the answer choices are a bit spread out -- we can count the multiples of 33 simply by dividing 33 into 900:
900/33 = 300/11 = a bit more than 27.
Thus, there are 27 multiples of 33 between 101 and 1000.
The correct answer is B.”

“OA: A neutron star, as small as 12 miles in diameter but containing half as much matter as the Sun, has a gravitational force at its surface about 67 billion times greater than that on Earth.
The green portion conveys the following meaning:
The words in brackets are omitted so that the green ...”

“I believe that the problem has been transcribed incorrectly.
Statement 1 should indicate that Rasheed bought not 2 fewer packages but only 1 fewer package, as shown in the following screen shot:
https://i.postimg.cc/t7bcm77Z/rasheed_Gprep-ds-2.jpg
If we know the number of chocolate BARS that ...”

“In the number line shown, the distance between 0 and a, a and b, and a and c is in the ratio of 1:2:3.
To determine the ratio of a to b to c, TEST AN EASY CASE.
Let a=1.
Since a is ONE PLACE to the right of 0, b must be TWO PLACES to the right of a (implying that b=3), while c must be THREE ...”

“Let M = the rate for each man and W = the rate for each woman.
The TIME RATIO for 12 men and 16 women to 13 men and 24 women is 5 days to 4 days.
Since time and rate are RECIPROCALS, the RATE RATIO for 12M+16W and 13M+24W is equal to the reciprocal of the time ratio:
(12M+16W)/(13M+24W) = ...”

“When a value increases repeatedly by r%:
Final amount = Original amount * (1 + r/100)^number of changes.
Let original amount = 1.
Since the original amount increases by a factor of 16:
Final amount = 16.
Since the investment increases by 2% every quarter:
r = 2.
Since x = the number of ...”

“We can PLUG IN THE ANSWERS, which represent the amount of cider.
When the correct answer is plugged in, the amount of alcohol in the 750 ml bottle = 87 ml.
Since the alcohol percentage in the cider = 5% = 1/20, the correct answer is almost
certain to be a multiple of 20.
Of the five answer ...”

“Average = sum/quantity.
In this case:
Average speed = (sum of the minutes spent at each speed)/(total number of minutes).
Total time spent at 15mph = 30 minutes.
Total time spent delivering pizzas at 0mph = 30 minutes.
Total time spent at 30mph = 15 minutes.
Average speed = (15*30 + 0*30 + ...”

“P(spoiled apple is selected) = 1 - P(spoiled apple is NOT selected).
P(1st apple is not spoiled) = 4/5. (Of the 5 apples, 4 are not spoiled).
P(2nd apple is not spoiled) = 3/4. (Of the 4 remaining apples, 3 are not spoiled).
Since we want both events to happen, we multiply the fractions:
...”

“Statements combined:
Each statement conveys the same information: that the average speed for the whole trip was 5 miles per hour.
The AVERAGE speed for the whole trip does not constrain the speed AT THE HALWAY POINT.
At the halfway point, the speed could be ANY NONNEGATIVE VALUE.
Thus, the ...”

“The answer choices represent the ratio between the numerator and the denominator of the given expression.
Option A implies that the numerator is about 14 times the denominator
Option B implies that the numerator is less than 7 times the denominator.
Option C implies that the numerator is 41 times ...”

“Since the answer choices are very spread out, we can BALLPARK -- and we can be somewhat lax with our estimations.
Numerator:
If the 11 greatest terms were 10²=100, the sum of the 11 greatest terms = 11*100 = 1100.
If the 11 greatest terms were 20²=400, the sum of the 11 greatest terms = ...”

“The sum of the 3 angles of a triangle = 180.
The sum of the parts of the ratio = 3+4+5 = 12.
Since 180/12 = 15, the multiplier for the ratio is 15, implying that the largest angle = 5*15 = 75.
The correct answer is D.”

“Statement 1:
Case 1: 1 sweater is sold for $270
Case 2: 3 sweaters are sold for $90 each
Since the number of sweaters sold can be different values, INSUFFICIENT.
Statement 2:
Since the $30 markup is 1/2 the cost of each sweater, the cost of each sweater = 60.
Thus, the selling price of ...”

“The GRE is administered by ETS, which also used to administer the GMAT.
The GRE Quantitative Guide states that any side of a triangle or parallelogram can be used as a base.
The GMAT is likely to abide by this definition.
But the terms length and long seem to connote a different meaning when ...”

“Since Doris is 12 years older than Fred, let Fred = 2 and Doris = 2+12 = 14.
y = sum of their ages = 2+14 = 16.
Since Fred = 2 and y=16, Fred''s age y years from now = 2+16 = 18. This is our target.
Now plug y=16 into the answers to see which yields the target value of 18.
Only D works:
3y/2 - ...”

“The trick is translate the chemical relationship into the following equation:
R = A²/B
Here''s why the equation above works:
1)The problem states that R is directly proportional to A². Directly proportional means that as one value increases, the other value also increases by a ...”

“Of the 50 fish caught in the second catch, 2 were tagged:
2/50 = 4/100 = 4%.
Since the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond, the total number of tagged fish -- 50 -- must be equal to about 4% of the total number of fish in the pond:
50 = ...”

“Since m=2 will yield an integer value for the given expression and will make the math easier, ignore the condition that m>2 and let m=2.
Plugging m=2 into 5^(m-2) * 2^(m+2), we get:
5⁰2⁴ = 1*16 = 16.
Since the result is a 2-digit integer, n=2.
The correct answer must work for n=m=2.
Only ...”

“Let the rate for each man = 1 widget per day, implying that the rate for 24 men = 24 widgets per day.
Since 24 men complete the job in 16 days, the total amount of work = rt = 24*16 widgets.
Since 32 women require 24 days, the rate for 32 women = w/t = (24*16)/24 = 16 widgets per day.
Since ...”

“Given that the answers choices are integers, I believe that the following reflects the problem''s intent:
The three fractions in the prompt -- 1/4, 1/3 and 1/2 -- imply that the fine must be divisible by 4, 3 and 2.
Smallest possible case:
Total fine = 4*3*2 = 24.
Since Sam paid $4 less ...”

“Option B for SC18 in the OG12:
Plants are more efficient at acquiring carbon than fungi.
Logical interpretation 1:
Plants are more efficient at acquiring carbon than
Logical interpretation 2:
Plants are more efficient at acquiring carbon than fungi
Since B allows for more than one logical ...”

“The alarm sounds for 3 of EVERY 4 BEARS that actually appear,
If 4 bears appear, then the number of alarms = 3.
Since the alarm sounds only 3 times for 4 bears, 1 of these 4 bears is undetected.
Since there are 10 false alarms for every 1 undetected bear, the number of false alarms = 10.
Thus, ...”

“The answer choices represent the price per pound:
(total cost)/(total number of pounds).
Since 7 pounds are purchased, the correct answer must have a denominator of 7.
A, C and D lack denominators and thus can be eliminated.
Since the denominator in B does not have to be 7, eliminate B.
The ...”

“Statement 1:
Time and rate have a RECIPROCAL relationship.
3 TIMES AS FAST implies 1/3 THE TIME.
Thus, the 2 hours that would be spent traveling 3 times as fast must be equal to 1/3 the actual time:
2 = (1/3)t
t = 6 hours.
SUFFICIENT.
Statement 2:
Here, the time can be any nonnegative ...”